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7.3 Gravitational Potential Energy (symbol: U and units: J)

7.3.1 in Work, Energy & Power topic

In the topic of Work, Energy & Power, the calculation of gravitational potential energy (GPE) is determined using the expression, mgh, which assumes that the gravitational field strength g is roughly constant, but a more robust formula is required for varying field strength, say when sending rockets which travel into outer space from the surface of the Earth.

ߡU = ߡPE = mgh or mgߡh

ΔU = ΔPE = mgh or mgΔh

where ΔPE or ΔU is the gravitational potential energy possessed by mass m measured from the reference height h0 , g is the gravitational field strength created by source mass M.

h or Δh is the change in the height (h1 - h0 ) of the mass m.

7.3.2 more universal formula for Potential energy

This formula PE = mgh or mgΔh was appropriate when the magnitude of gravitational field strength, g of Earth was assumed to be constant (9.81 m s-2) over this height. When the assumption of constant gravitational field strength is not valid, a more robust PE formula is given by the calculation of potential energy 

U = -mG M r


U is the gravitational potential energy possessed by mass m measured from the reference position of infinity,

G is the gravitational constant, approximately 6.673×10−11 N·(m/kg)2,

M is the source mass that sets up the gravitational field,

m is the test mass experiencing the gravitational field set up by M, and

r is the distance away from the centre of the gravitational field source mass M

 U = -mG M/ r

In this particular case, U = -m G M r = -(1) 6.67 x 1 0 -11 ( 100) 100 = - 6.67 x 1 0 -11 J

Calculate for the case when r = 50 m.

 U = -mG M/ r

Similarly, U = -m G M r = -(1) 6.67 x 1 0 -11 ( 100 ) 50 = - 13.35 x 1 0 -11 J

7.3.3 Characteristics of Gravitational potential energy

equation is given by  U = -m G M r

a scalar quantity (i.e. it has no direction only magnitude )

has only negative value due to the reference potential equal zero U = 0, position is defined at position r = infinity,  ∞, and gravity is attractive which means the lowest potential position is at the centre of the source mass M.

U = -infinity, at r = 0, lowest potential energy

U = 0, at r = infinity, highest potential energy   

Now, this leads to the need to define gravitational potential energy.

7.3.4 Definition of gravitational potential energy, U

The gravitational potential energy (U) of a mass m at a point (due to the gravitational field set up by mass M ) is defined as the work done by an external agent in bringing the mass from infinity to that point r distance away from source mass M.

To further understand the meaning of this definition, let's start with the definition for work done on external agent, where WD = WD by external agent.

W D = r F d x
Substituting the formula for force F and working out the integration, it can be shown.

W D = r F d r = r G M m r 2 dr= [ - G m M r ] r = [ - G m M r - - G m M ] = - G m M r

Notice the formula for this work done by external agent is equal to the U.

W D = U = - G m M r

To understand this definition, we can use the simulation with some imagination.
Imagine the test mass, m is at a distance r = ∞, U = 0 (the simulation does not allow r = ∞, but let's assume r = 100 is somewhat far enough)

Let imagine a force pointing towards infinity away from source mass M, acting on this test mass m, the potential energy of m at this point would be therefore the work done by this external force.

WD= Ufinal-Uinitial=Ur-U

which means that the work done by external force is equal to the gravitational potential energy (Ur) since U = 0.

7.3.5 Suggested activity

The intent of these activities is to allow students be like a student-scientists and collect their own data to form an understanding of potential energy.

The test mass (green) is at r = 100 m, observe and record the value for U r=100 = -6.67 x10-11 J

Click play and pause the model when r = 90 m, record the value of U r=90 = ____________ J

Click play and pause the model when r = 80 m, record the value of U r=80 = ____________ J

Click play and pause the model when r = 70 m, record the value of U r=70 = ____________ J

Click play and pause the model when r = 60 m, record the value of U r=60 = ____________ J

Click play and pause the model when r = 50 m, record the value of U r=50 = ____________ J

Click play and pause the model when r = 40 m, record the value of U r=40 = ____________ J

Click play and pause the model when r = 30 m, record the value of U r=30 = ____________ J

Click play and pause the model when r = 20 m, record the value of U r=20 = ____________ J

Calculate the change in U from r = 100 to r = 50.

Answer is about ΔU = - [ 6.67 x 1 0 -11 ( 1 ) ( 100 ) 50 - 6.67 x 1 0 -11 ( 1 ) ( 100 ) 100 ] = - 6.67 x 1 0 -9 J

Check the "your model" checkbox and a teal color line appear. For example if the model is ,  U = - G m M r , key in -6.67*1*100/abs(r) and observe the closeness of fit of the orange color line (data collected) versus the teal color line (model proposed).

Suggest with reason why you think the model -6.67*1*100/abs(r) is accurate.

Note that the model already has a multiple of x10-11 so there is no need to key it.

7.3.6 Model:

  1. Run Sim


Code Language Translator Run

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This email address is being protected from spambots. You need JavaScript enabled to view it.; Anne Cox; Wolfgang Christian; Francisco Esquembre

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  • Secondary
  • Gravity
  • Chemistry
  • Chemistry of Reactions
  • Junior College
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